\(\int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx\) [118]

Optimal result

Integrand size = 11, antiderivative size = 19 \[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=\frac {\log (a+b \text {csch}(x))}{a}+\frac {\log (\sinh (x))}{a} \]

[Out]

ln(a+b*csch(x))/a+ln(sinh(x))/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3970, 36, 29, 31} \[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=\frac {\log (a+b \text {csch}(x))}{a}+\frac {\log (\sinh (x))}{a} \]

[In]

Int[Coth[x]/(a + b*Csch[x]),x]

[Out]

Log[a + b*Csch[x]]/a + Log[Sinh[x]]/a

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 3970

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \text {csch}(x)\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,b \text {csch}(x)\right )}{a}+\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \text {csch}(x)\right )}{a} \\ & = \frac {\log (a+b \text {csch}(x))}{a}+\frac {\log (\sinh (x))}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=\frac {\log (b+a \sinh (x))}{a} \]

[In]

Integrate[Coth[x]/(a + b*Csch[x]),x]

[Out]

Log[b + a*Sinh[x]]/a

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(-\frac {\ln \left (\operatorname {csch}\left (x \right )\right )}{a}+\frac {\ln \left (a +b \,\operatorname {csch}\left (x \right )\right )}{a}\)	\(21\)
default	\(-\frac {\ln \left (\operatorname {csch}\left (x \right )\right )}{a}+\frac {\ln \left (a +b \,\operatorname {csch}\left (x \right )\right )}{a}\)	\(21\)
risch	\(-\frac {x}{a}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a}\)	\(27\)

[In]

int(coth(x)/(a+b*csch(x)),x,method=_RETURNVERBOSE)

[Out]

-1/a*ln(csch(x))+ln(a+b*csch(x))/a

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=-\frac {x - \log \left (\frac {2 \, {\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \]

[In]

integrate(coth(x)/(a+b*csch(x)),x, algorithm="fricas")

[Out]

-(x - log(2*(a*sinh(x) + b)/(cosh(x) - sinh(x))))/a

Sympy [F]

\[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth {\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]

[In]

integrate(coth(x)/(a+b*csch(x)),x)

[Out]

Integral(coth(x)/(a + b*csch(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} + \frac {\log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a} \]

[In]

integrate(coth(x)/(a+b*csch(x)),x, algorithm="maxima")

[Out]

x/a + log(-2*b*e^(-x) + a*e^(-2*x) - a)/a

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=\frac {\log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a} \]

[In]

integrate(coth(x)/(a+b*csch(x)),x, algorithm="giac")

[Out]

log(abs(-a*(e^(-x) - e^x) + 2*b))/a

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {\coth (x)}{a+b \text {csch}(x)} \, dx=-\frac {x-\ln \left (2\,b\,{\mathrm {e}}^x-a+a\,{\mathrm {e}}^{2\,x}\right )}{a} \]

[In]

int(coth(x)/(a + b/sinh(x)),x)

[Out]

-(x - log(2*b*exp(x) - a + a*exp(2*x)))/a